What is a dimension?

[abskebabs]

I feel that I could have asked this question in either the physics or mathematics part of this forum, but as the context of my question is very general, I decided to place it here. We often talk about dimensions in different circumstances and situations describing different phenomena, both abstract and "physical". My question is there a rigorous single definition for what we call a dimension? If so what should it be?

 

For example in physics we refer to spatial dimensions, and time as dimensions, and these seem to be because of geometric relations that unify them which are not trivial. One only needs to mention the spacetime invariant interval, or Pythagorean theorem to recognise this.

 

However, in my brief knowledge I am also aware of different notions of dimensionality, such as similarity dimensionality, as well as many others that can be applied in a useful way when analysing fractals. These are also defined in a way that is self consistent.

 

As well as this, I know of, or more accurately am vaguely aware of other types of dimensionality like those in things like Hilbert spaces etc. Can all these be "reconciled", so as to provide us with a single rigorous definition of dimensionality? Or is this a mistake, and not feasible or worth doing; and we should recognise the inherent differences between the notions, and perhaps it is just a mistake of our intuition mixed with our everyday experiences and notions to expect there to be an underlying connection. I expect my questions may sound a little naive to those more seasoned with the topic under discussion, but oh well:cool:

[Bignose]

This is admittedly going to be very non-specific and non-rigorous like completely formal mathematics can be. But, my understanding of dimension has always been that a separate dimension is needed for any quantity that cannot be described by any other dimensions.

 

That's worded a little oddly, so, let me give you some examples:

 

You cannot describe depth in any way whatsoever using only length and width, or, to put it another way, you cannot describe a point's position in z using only x and y. Unless there is a known relationship between x,y,& z, like tracing a curve or edge of a solid.

 

Or another example, you cannot describe the velocity of a particle using only it's position. Or (taken from an example of my engineering work) a particle's size, or shape, or reactivity, or porosity, or concentration using only its position and/or velocity. Each of those requires using another variable, which can be thought of as another dimension, to describe the distribution of particles. One of the first examples of using something like this is the kinetic theory of gases, where they integrate over the 6 dimensional space of position and velocity to describe a population of gas molecules.

 

So, again, the definition isn't very formal at all, but that's how I always thought of dimension -- it's a quantity that cannot be described using any of the other dimensions.

[Dave]

However, in my brief knowledge I am also aware of different notions of dimensionality, such as similarity dimensionality, as well as many others that can be applied in a useful way when analysing fractals. These are also defined in a way that is self consistent.

 

When first reading this post, I'd originally intended to write quite a long reply as it's a very interesting question. But the more I thought about it, the more I've realised that I actually have very little understanding of how to discuss this kind of thing. But I'll try to sum up my thoughts here.

 

Let's consider the vector space [math]\mathbb{R}^n[/math] (under the operation of addition). Then, since we can span the space by precisely [math]n[/math] vectors, we say that it has dimension n. Intuitively, this makes sense: one can imagine n different degrees of freedom.

 

Edit: This is rubbish, ignore

 

Now, I'm just going to assume you know what both the box dimension and Hausdorff dimension are. If you don't, you'll need to look then up. But rest assured, they are very sensibly defined, and work quite well. For instance, on the space [math][0,1] = \{ x \in \mathbb{R} \ | \ 0 \leq x \leq 1 \}[/math] we have that [math]\text{dim}_{\text{B}} [0,1] = \text{dim}_{\text{H}} [0,1] = 1[/math], which again, makes sense - one degree of freedom.

 

But when we apply this to [math]\mathbb{R}^n[/math], we find [math]\text{dim}_{\text{B}} (\mathbb{R}^n) = \text{dim}_{\text{H}} (\mathbb{R}^n) = \infty[/math]. This doesn't really tally with what we did before - quite a glaring inconsistency, in fact.

 

This is, as far as I know, correct.

 

Not only this, but indeed the box/Hausdorff dimensions do not agree for particular sets. For instance, the set [imath]G = \{ 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots \}[/imath] has [imath] \text{dim}_{\text{H}} \, G = 0[/imath] (because it is countable) and [imath]\text{dim}_{\text{B}} \, G = \frac{1}{2}[/imath].

 

Don't really know what else to say, so I'll end my response here. I'll try to answer questions you have though

 

So, again, the definition isn't very formal at all, but that's how I always thought of dimension -- it's a quantity that cannot be described using any of the other dimensions.

 

That's a pretty good intuitive interpretation of dimension. (Essentially what I outlined in the post above).

 

However, consider something like the Cantor set. At first glance, it's a set which has Lebesgue measure 0; this should indicate that it's essentially a set of 'points' (something like [math]\{0, 1/2, 3/4, \dots\}[/math]), so should have dimension 0, the same as a point. Indeed, the Cantor set is totally disconnected.

 

But then you take a closer look: every point in the set is an accumulation point. That is, if we're at a point in the Cantor set and look in any (small) neighbourhood, we'll find at least one other point there. In fact, the space is uncountable.

 

So it essentially boils down to this: is it sensible to define a notion of dimension where an uncountable set is allocated dimension 0? In my mind, probably not. Certainly the dimension should be < 1, but in my mind it should also be > 0. Under both the Hausdorff and box dimension we actually have the dimension being [math]\log 2/\log 3[/math], which seems reasonably sensible.

 

Anyway, this isn't a criticism, just trying to highlight how difficult identifying a sensible notion for dimension actually is.

[gib65]

I've always thought (and my thoughts are very layman-like) that a dimension could be described as having the properties of linearity, continuity, and infinit extension in at least one direction. There's also how it relates to other dimensions of its kind, and that is that whatever the dimension is a scale or measure of (distance, energy, time, etc.), things that find a place on that dimension must be able to sustain or vary that place independently of its sustaining or varying its place on other orthogonal dimensions. For example, if you take one spatial dimension (call it the x-axis) and find a particle to be placed at a specific point on that dimension (say at x=5), that position should stay fixed at 5 or move away from 5 independently of its position on any other spatial dimension (call them the y-axis and z-axis) whether fixed or varying.

 

This understanding of a dimension works just as well with physical phenomena (like space and time) as it does with non-physical or abstract phenomena, such as personality traits. For example, the theory in psychology called "the big-5" theory of personality has it that personality can be broken down into 5 major factors (Neuroticism, Extraversion, Agreeableness, Conscientiousness, and Openness to Experiences). These can be considered dimensions of personality since, theoretically, the personality dispositions of any one random person varies along each of these personality traits independently of any of the other four traits (although I think some studies show the big-5 to be not completely independent of each other, but I'm not sure).

[Royston]

Just to echo what Bignose stated, that's how I've interpreted dimensions.

 

It's interesting how dimensions seem (to me at least) essentially more abstract when they're whittled down, which is the first use of dimensions anybody comes across i.e you learn the relationship between two variables, say, using differentiation, and you start to realize how dimensions are 'built up' for want of a better term.

 

This was actually a thought I posed to my house mate, the other day, but as soon as I thought I had a clear idea on how to explain dimensions, I realized how mathematical the word really is...i.e an understanding of the terms would require practicing some maths, which, at the time, would of been problematic

[someguy]

This probably won't be new to you, but for me there are 2 sorts of answers. the first i would say is simply mathematical. the drawing of graphs. one constant i guess would be a point. 6=6... 6 basically zero dimensions. one infinitely tiny one, not really one at all. one variable would be a line, x=6, one dimension two a plane, two dimensions, 3 3 dimensions, 4,4 and so on. that is how i would define dimensions in a general way.

 

the other way to answer it would be what one dimension is, what 2 dimensions are, 3, and 4. but as we can see, at least from our dimensional construction and perspective on the 4 dimensions, the 4th is fundamentally different from the other 3, so in that way i can't see a definition that can encompass all dimensions. unless you go the math way, and probably you could find a better one than the previous example. i have no idea what the 5th dimension is like, and you could propose any ludicrous idea and i could only answer with, maybe it is that way. but to me it doesn't much matter since i am not yet convinced that a 5th exists, but i expect if ever i do i will also be bale to know what it is. so in short i think for a non math definition of dimension you would need to define them one by one. maybe someone else has some better ideas, that's a good question.

[pioneer]

When you do work with crystals, sometimes it is easier to describe them with more than three spatial dimensions, say (a,b,c,d), since the planes of some crystals have four basic directions. Inspite of using 4-D to describe the crystal, the crystal still exists in 3-D space. The 4-D is done primarily to make the math easier and has nothing to do with there being more that 3 spatial dimensions within the crystals. One could still do it with (x,y,z) but the math gets too cumbersome. This conversion makes it easier but can lead to confusion when people assume this extra dimension in crystals gives them magically healing properties.

 

If I wanted to plot flowers as a function of white, red, pink, blue and yellow I could use a 6-axis plot, to put everything on the same graph for easy cross reference. But that does not mean this flower exists in 6-D. It still exists in 3-D but the properties of color are easier to express with 6-D. If I was clever, I could use the flower as t, and use the three primary colors as x,y,z .That way I could reduce my plot to 4-D, which is reality.

 

Dimension in a loose sense is a place to put each variable to make the math easier. But this should not be confused with space-time, which is 4-D. The more dimensions we have, the easier it is to include everything. If computers did not exist, these n-D graphs would get confusing, and there would be a push to increase the number of graphs so each will only have maybe 3-D. Then the results of these many graphs would be replotted also in 3-D, etc., until the final plot looks like the 4-D of space-time. This way science would circumvent speculation about magical places.

[Dave]

I'd just like to point out that, since this was posted on the mathematics fora, I would ask that any discussion of the topic remain solely on mathematical interpretations of the word (i.e. not physically related).

[abskebabs]

To be honest with you Dave, I have only recently(i.e. this year) come to understand the meaning of similarity dimensionality, and that itself was something short of a revelation, because it exposed to me how little I had thought of the notion dimension properly before(In fact I would recommend people unfamiliar with it to look it up as it's fairly simple, and could give you some real insight). It's amazing really. I may read up more on Haussdorff and box dimensionality in the near future(as at the moment I consider myself only very vaguely familiar with them) to give myself a better idea, and so then I can ask you questions a little more specific and perhaps less redundant.

 

When our lecturer went through similarity dimensionality, it really struck out for me how we could take a notion in how we define the conventional Euclidean 1d to 3d spaces, and then abstract our notions to other self repeating pictures for lack of a better word. Anyway, I thank everyone for taking an interest. If I don't forget about this thread, I may come back to it later.

[Martin]

 

Now, I'm just going to assume you know what both the box dimension and Hausdorff dimension are. If you don't, you'll need to look then up. But rest assured, they are very sensibly defined, and work quite well. ...

 

But when we apply this to [math]\mathbb{R}^n[/math], we find [math]\text{dim}_{\text{B}} (\mathbb{R}^n) = \text{dim}_{\text{H}} (\mathbb{R}^n) = \infty[/math].

This doesn't really tally...

...

 

If that were true it would be really confusing!

But that might be a case of mixing up something called Hausdorff measure with the actual Hausdorff dimension which is defined using the measure, and is better behaved than that.

 

A possible place to look up Hausdorff dimension is Wikipedia

and there one learns that

 

[math]\text{dim}_{\text{H}}\mathbb{R}^n = n[/math]

 

http://en.wikipedia.org/wiki/Hausdorff_dimension

 

In other words the Haus. dim. of ordinary Euclidean R³ is 3, after all, just as one expects.

 

... I have only recently(i.e. this year) come to understand the meaning of similarity dimensionality,...

 

I find that one intuitive. Wolfram has an online definition

http://mathworld.wolfram.com/SimilarityDimension.html

 

Sim. dim. is a lot easier than Hausdorff to grasp and apply in simple situations, IMO, which doesnt mean it is better. Probably Hausdorff himself started at square one with the similarity idea and found he needed to beef it up to handle some hairy special cases.

 

Let's take the simplest case where you just want to find the dimensionality of the space you live in around some point. ( the complete definition talks about the dimensionality of objects or shapes. )

 

With sim-dim you just take a ball around that point and see how the volume grows as you increase the radius. If the volume increases as the 3rd power of the radius then you know you are living in a 3D space.

 

I like the direct experiential quality. It makes dimension an OBSERVABLE. If you forget what the dimension is of our universe you can do an EXPERIMENT to find out. Just inflate a sphere and measure the volume and see whether the volume grows as the square of the radius, or the 2.5 power or the 4th power or whatever. If it grows as the 3rd power everything is all right.

 

If you ever find yourself in a different universe then even if the lights are off you can tell the dimension by comparing radius and volumes of things.

 

I guess that is what your teacher was talking about Abs. Or was he talking about something else?

[Dave]

Man, I don't know what I was thinking when I wrote that stuff about Hausdorff dimension. It's not like I took a course on it or anything Hopefully the stuff I wrote about the Cantor set won't be a load of twaddle either, otherwise I should just shoot myself now.

 

Edit: Annoyed at myself for being such an idiot, so thought I'd explain why I got so confused. Aside from the fact I haven't touched it in a while, the Hausdorff dimension relies on the s-dimensional Hausdorff measure [imath]\mathcal{H}[/imath] to actually determine the dimension (namely, [math]s = \text{dim}_{\text{H}}(G) \Leftrightarrow s = \inf \{ t > 0 \ | \ \mathcal{H}^t (G) = \infty \}[/math]). I plucked infinity out of the air from this, rather stupidly.

[Martin]

no need for chagrin

technical definitions aren't holy

and I'm glad you jumped in and responded to Abs question immediately because it made a nice discussion thread.

we are spread thin. I rarely rarely look in at math, because intensely interested in physics these days.

 

BTW dimension is a case where there is no clear boundary between math and physics concerns.

 

In one of the approaches to quantum geometry/gravity that have sprung up in past ten years, both the spatial and the spacetime dimension are quantum observables!

 

this is work at Utrecht (at 't Hooft's institute) by Ambjorn and Loll and postdocs.

they generate simulation universes in the computer (actually millions of universes!) and explore them-----do things like measure the spatial dimensionality at many sample points in the universe----and then roll the dice again and move on.

 

they use two main methods to observe the dimension in one of their MonteCarlo universes. One is what Abs would call the "similarity dimension" (or maybe there's a better term like "ball dimension") where you just take the ball around the point---the set of things within R steps---and see how volume grows with radius. the other method they use is called SPECTRAL dimension, and you measure it by running a random walk starting at the point and empirically determining the probability of getting lost or of accidentally returning. In higher dimension you have a higher probability of never returning to start.

 

One of Ambjorn and Loll triumphs is that their quantum geometry can now produce universes that exhibit 3D spatial and 4D spacetime at macroscopic scale even though the universes assemble themselves and evolve according to quantum rules and are "foamy" and even "fractal-like" at very small scale. At microscopic scale even the dimensionality behaves oddly and does not have to be a whole number. It might be, like 1.9 or 2.5 instead of exactly 3 or exactly 4.

 

I think their work is beautiful both as QG physics, but also in a sense as mathematics. exploring a really new kind of continuum

 

glad we have a thread about dimension

[abskebabs]

Martin, for your benefit, and especially for people unfamiliar with the subject matter, I thought I would outline what I understand by similarity dimensionality.

 

Suppose we start off with a line of length 1. We can split it up into N equal pieces each of length 1/N. Now if we represent(arbitrarily in this case) the number of pieces by the letter k, and the inverse proportion of their length(basically 1/length) by say, F, then we have by similarity dimensionality:

[math]k=F^d[/math] Where d represents the number of dimensions

In our case however, F=N, and K=N; so the line is 1 dimensional.

 

We can take this same process and now apply it to a cube. Along the 3 connecting edges at a corner of a cube of length 1 suppose we split the lengths of the edges into N different segments, and make marks to represent their boundaries. If we imagine cutting the cube straight along these marks, we end up splitting it into [math]N^3[/math] pieces. Having done this, we see that now; [math]K=N^3[/math] and[math]F=N[/math]. Therefore we have:

[math]N^3=N^d[/math] so we don't have to take logs to see that the number of dimensions is 3.

 

Now that I've established this, I will show how it can be used to look at the dimensionality of a more solely mathematical, but simple object, namely: the Garnett set. Now the Garnett set is the union of all that is covered by the following operation carried on infinitely on a square:

Take a square of length 1(to start off with), and take from it 4 squares at each of the corners, each square of length 1/4. For the next step I would take each of the 1/length squares and carry out the same procedure. This process would then be carried out to infinity as indicated above. Therefore after doing this k times I would have[math]4^k[/math] pieces each with a length [math]4^{-k}[/math].

 

Just like the ordinary cases I have already described above we can define the self similarity for the Garnett set just as well. We notice at each stage the square is split into 4 more pieces each of length a 1/4 of the square from which they were constructed. Therefore we have k=4 and N=4!

So; [math]d=\frac{ln{4}}{ln{4}}[/math] therefore d=1.

The set is 1 dimensional in terms of self similarity. Now this kind of result I would never have intuitively guessed if someone asked me about dimensionality. I was thinking a lot when I first saw the results, and I guess I came to terms with it when I faced up to the fact that when we define a notion of dimensionality like that involving self similarity, we can then import that and apply it to a wide range of more abstract objects or phenomenon.

 

But I didn't see THIS type of dimensionality necessarily having any physical significance(especially with regard to what I have referred to as more abstract objects), and I guess I feebly attempted to suggest my feelings about this before now.

 

Martin, I notice you gave the example of a sphere when defining a notion for self similarity. For my own satisfaction, I think I will just try to find or confirm its dimensionality within this thread:

Ok, so we have the radial length of say 1. Divide along the length into N equal portions. The volume of the sphere will be [math]\frac{4}{3}\pi*[/math], so we should have [math]\frac{4}{3}\pi*N^3[/math] "pieces", however I recognise that this would only be true if we were talking about how many cubes of length 1/N would fit inside the sphere. I suspect I would get the result you no doubt expected if I tried to divide the sphere into spheres of size [math]\frac{4}{3}\pi*\frac{1}{N^3}[/math], as [math]N^3[/math] of these spheres would produce an equivalent volume. But conceptually, I see no sensible notion for how to subdivide the sphere into these "mini-spheres", as wouldn't it be impossible(at least if they were to maintain their shape, and hence self similarity)? For example, I can imagine(me and my interesting life:-p ) trying to fill a large sphere with [math]N^3[/math] smaller spheres each of radius [math]\frac{1}{N}[/math] of the large sphere's radius, but this would cause it to "overflow", and I wouldn't be able to completely fill the large sphere with them because of the inherent gaps that appear.

 

Therefore, I think I can kind of give you an idea after this terribly long post why I don't think the notion of self similarity dimensionality can be as easily defined self consistently, at least on an elementary level for spheres as it can be on cubes. Instead, I feel the notion could be "abstracted", to give the expected conclusion(that the sphere is 3d). Please tell me your thoughts on this. Phew!